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Sketching the Gradient Function from a Graph

Given the graph of a function $y = f(x)$ we can sketch the gradient function denoted by $f'(x)$.

The gradient function of a polynomial is one degree less than the original function.

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To sketch a gradient function consider each section of the original graph

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Conditions of differentiability

Recall (from Conditions of Differentiability) that the gradient function or derivative, $f'(x)$ exists for a given value of x only if the graph of f(x) is smooth and continuous at that x-value. It must be possible to draw a unique tangent at that x-value.

This means that the gradient function (derivative) does not exist (is not defined) in the following conditions:

where there is a vertical asymptote

where there is a break or a hole

where there is a sharp corner (cusp)

at the endpoints of the domain

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Gradient Functions Questions

For each of the functions shown, sketch the graph of its gradient function. State the domain of each gradient function.